Following the works of Willis [150, 152], in this subsection we deal with the sub-structural char- acterization of a random, single-scale, granular composite. To this end, we consider the material system of Fig. 2.1(a), which is made out of different grains distributed randomly over the RVE Ω. For the purposes of the current discussion, each grain of Fig. 2.1(a) is assumed to behomogeneous, so that the local material properties vary from grain-to-grain but not within the individual grains (i.e., we ignore part (b) of this figure). The term family (orphase) is used here for the set of all grains exhibiting identical constitutive behavior (i.e., all grains shown in Fig. 2.1(a) with a specific color). For the purpose of generality, the RVE Ω of Fig. 2.1(a) is taken to be made out of an arbitrary number 𝑁 of grain-families and each grain-family𝑟(𝑟= 1, ..., 𝑁) is assumed to occupy the subregion Ω(𝑟)of Ω. Formally, the sub-structure of thespecificcomposite under consideration could be prescribed through the characteristic functions
𝜒(𝑟)(x) = ⎧ ⎨ ⎩ 1, if x∈Ω(𝑟) 0, otherwise . (2.1)
In practice, however, for a random composite it is neither possible nor useful to know the char- acteristic functions𝜒(𝑟), as defined by (2.1), exactly. This point will become more evident in the discussion that follows. We note in passing that in the case of a periodic composite it is conceivable
for such information to be available, since it would only require knowledge of these functions over appropriately defined unit cells.
The notion of a random composite refers to a member of a sample space 𝑆 with a specific realization of the sub-structure (such as the granular system of Fig. 2.1(a)), labeled 𝛼for short, which is associated with a probability density 𝑝(𝛼) defined over𝑆. Therefore, the characteristic functions 𝜒(𝑟), as defined by (2.1), for a random system depend also on 𝛼. The relevant sub- structural information for such systems is naturally available in terms of the associated 𝑛-point probability functions. The one-point probability 𝑝(𝑟)(x) of finding a grain of type 𝑟at pointx is defined as the ensemble average
𝑝(𝑟)(x) =∫
𝑆
𝜒(𝑟)(x, 𝛼)𝑝(𝛼)𝑑𝛼. (2.2) The two-point probability𝑝(𝑟𝑠)(x,x′) of finding simultaneously a grain of type𝑟atxand a grain
of type𝑠atx′ is given by
𝑝(𝑟𝑠)(x,x′) =
∫
𝑆
𝜒(𝑟)(x, 𝛼)𝜒(𝑠)(x′, 𝛼)𝑝(𝛼)𝑑𝛼. (2.3) Higher-point probabilities are defined similarly. We emphasize that the sub-structural character- ization given by expressions such as (2.2) and (2.3) is valid for random single-scale composites of any type. For granular systems, in particular, such a description is well-suited since it allows consideration of multi-point statistical information for the positions of the grains and at the same time requires no specific grain-shape commitment.
At this point it is relevant to introduce the “separation of the length-scales hypothesis”, which is of fundamental importance in homogenization theory. Referring to Fig. 2.1(a), there are two important length-scales that are associated with the RVE Ω: (𝑖) the macro-scale 𝐿1, which is characteristic of the size of Ω and (𝑖𝑖) the length-scale ℓ1, characterizing the heterogeneity (i.e., the size of a typical grain). The separation of the length-scales hypothesis states that
ℓ1<< 𝐿1. (2.4)
The RVE Ω of Fig. 2.1(a) is assumed to be statistically uniform, i.e., the 𝑛-point probability functions are invariant under translations. This implies that the one-point probabilities 𝑝(𝑟)(x) are constants, the two-point probabilities 𝑝(𝑟𝑠)(x,x′) are functions of (x−x′), etc. Although, in
general, it is easy to think of heterogeneous systems (e.g., a periodic composite) in which these conditions fail, under the separation of the length-scales hypothesis (2.4) it is reasonable to expect that the random system of Fig. 2.1(a) is statistically uniform, except in some “thin layer” along its boundary. Furthermore, we will make use of the ergodic assumption that the configurations in neighborhoods centered at a given point x in the sample occur with the same frequency as they occur in neighborhoods centered at various points in a single realization. This allows the
replacement of ensemble averages with the corresponding volume averages over the RVE Ω. For example, expressions (2.2) and (2.3) may be respectively replaced by
𝑝(𝑟)(x) = 1 ∣Ω∣ ∫ Ω 𝜒(𝑟)(x′′)𝑑x′′= ∣Ω (𝑟)∣ ∣Ω∣ ≡𝑐 (𝑟), (2.5)
where 𝑐(𝑟)denotes the volume fraction of the grain-family𝑟, and
𝑝(𝑟𝑠)(x−x′) = 1 ∣Ω∣
∫
Ω
𝜒(𝑟)(x+x′′)𝜒(𝑠)(x′+x′′)𝑑x′′. (2.6)
In addition, it will be assumed that the granular composite under consideration possessesno long- range order, which implies that
𝑝(𝑟𝑠)(x−x′)∼𝑝(𝑟)(x)𝑝(𝑠)(x′), (2.7) for large values of ∣x−x′∣. The meaning of this assumption is that the expectations of finding a
grain of the type 𝑟 at pointx and a grain of the type𝑠at point x′ are statistically independent
whenxandx′ are far apart. Notice that the separation of the length-scales hypothesis (2.4) is of
crucial importance for condition (2.7) as well.
For simplicity, in this work we will restrict attention to one- and two-point probability func- tions. Furthermore, we will make use of the “ellipsoidal symmetry” hypothesis for the two-point probabilities (due to Willis [150]), i.e., we will assume that𝑝(𝑟𝑠)depend on (x−x′) only through
the combination ∣Z(𝑑𝑟𝑠)(x−x′)∣, where Z(𝑟𝑠)
𝑑 are constant, symmetric, positive definite, second- order tensors. Note that, due to the property𝑝(𝑟𝑠)(z) =𝑝(𝑠𝑟)(−z) deduced from (2.6), the tensors
Z(𝑑𝑟𝑠) posses the symmetry property Z
(𝑟𝑠)
𝑑 = Z
(𝑠𝑟)
𝑑 . The special case Z
(𝑟𝑠)
𝑑 = I corresponds to the well-known “statistical isotropy” assumption, i.e., the two-point probabilities are orientation independent. More generally, the tensors Z(𝑑𝑟𝑠) define the following ellipsoids centered atx𝐶
Ω(𝑑𝑟𝑠)={x∣(x−x𝐶)⋅Z(𝑑𝑟𝑠)(x−x𝐶)≤1}, (2.8) which are characteristic for the distribution of the grains in the RVE Ω. Finally, it is remarked that in this work we will restrict consideration to granular systems such that Z(𝑑𝑟𝑠) ≡ Z for all pairs (𝑟𝑠), with 𝑟, 𝑠= 1, ..., 𝑁. This assumption is indicated in Fig. 2.1(a) by the doted ellipses (since only a cross section of the material is shown), which are shown to have the same shape and orientation for all grains.